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A Universe BuilT On Loops
An Introduction to Loop Quantum Gravity
Katharina Fließer
A Universe builT on Loops
Katharina Fließer
Abstract
In situations in which general relativity breaks down, for example at the beginning of our universe, a
theory of quantum gravity is expected to be necessary to describe the dynamics of space and time.
One of the best opportunities for such a theory at the moment is Loop Quantum Gravity. Loop
Quantum Gravity is a mathematically well-defined background-independent quantization of general
relativity. It is a field theory of gravity. Loop Quantum Gravity builds a spacetime out of so called
“loops”. The universe as it is today is described as a tangle of an uncountable number of such loops,
each of them bringing along one tiny piece of space. If we reverse the process, remove one loop after
the other, and every time remove a little bit of space too, we finally get to a point governed by a void
which can´t even be compared by the vacuum, because it isn´t empty space, but eventually no space
at all, a state of absolute emptiness.
Among the most significant results of Loop Quantum Gravity are the microstructure it provides
spacetime, and its ability to avoid spacetime-singularities like Black Holes and the Big BangSingularity itself. This, in particular, opens up the possibility of a theoretical investigation into the very
early universe, even to the beginning of our universe and most remarkable indeed spacetime-regions
before the Big Bang.
While general relativity is successfully used to describe the structure of space and time on large
scales[1], going to very small scales quantum effects become more and more significant. Most of the
universe as it is now isn´t influenced by quantum effects. It´s geometry is easily described by the
Einstein field equations of general relativity. But what if we take a closer look on the very early
universe? General relativity evolved backwards in time leads to the Big Bang Singularity, a point of
endless density where the whole mass of the universe is gathered. But this theory ignores the
quantum effects which affect and restrict the concentration of matter and the strange of gravity[2].
To correctly describe such circumstances, a quantum theory of gravity is needed.
One of the most successful quantum theories of gravity is Loop Quantum Gravity (LQG), which is kind
of a Quantum Field Theory (a theory in which a field is described by particles and their interactions)
of space and time. Loop Quantum Gravity shows, that if space and time become discrete, no more
singularities are needed to describe our universe and there might even be a time before the Big
Bang.
LQG started with a mathematical reformulation of general relativity. The purpose of this
reformulation was to find a mathematical description of gravity which could be compared with those
of the other forces (weak, strong and electromagnetic), so methods used in Quantum
Electrodynamics for example, could be used for gravity too. The three shorter ranged forces are
described by objects describing angular changes. In electromagnetism for example these objects are
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A Universe BuilT On Loops
An Introduction to Loop Quantum Gravity
Katharina Fließer
the electric and magnetic field lines. After rewriting general relativity, gravity could be characterized
by such objects (later called Ashtekar variables, after Abhay Ashtekar who first considered them) and
the area of the surface. A closed line in quantum gravity is called a loop which is the reason for the
theories name. The Ashtekar variables and the area of the surface do not commute, they underlie
Heisenbergs uncertainty principle. The consequence of this uncertainty is that whenever there is
another loop, there is also another quant of area, about the size of the Planck scale. That means that
space in Loop Quantum Gravity is build out of tiny particles, and if there aren´t any loops, there
won´t be any space at all. This point of few leads to a whole new description of spacetime[3].
A basic in Loop Gravity is the notion of “quantum of space”. It means that in the same way as a
photon is a quantum excitation of the electromagnetic field, there is a quantum excitation of the
gravitational field. But there is one important difference between those two cases: while the field in
Maxwell theory lives over a fixed space time, the other represents itself spacetime. A photon can be
localized in space, but the quanta of loop gravity define space itself and can only be localized in
respect to one another. The quantum numbers they carry (for a photon there would be momentum
and position) define a quantized theory as described in the spin-geometry theorem by Roger
Penrose, which is the root of LQG, describing a geometry with areas and angels. The quantum states
of space are
. Γ gives the adjacency relations between the quanta of space
gives the
area of the faces that separate two quanta of space and gives the volume of the quanta of
space. Those three states are called spin-network states. The theory they describe is discrete and
extremely similar to lattice gauge theory on a 3d lattice from Quantum Chromo Dynamics[4], the
Quantum Field Theory of quarks and gluons. There are even more cases in which LQG combines
the formal structures of Quantum Electrodynamics and Quantum Chromo Dynamics, which is a
beautiful feature of LQG, but here I haven´t got the space to illustrate more of it.
The equations describing the dynamics of space and time in LQG are the Quantum-EinsteinEquations:
̂
̂̂
̂
̂
an adaption of the Einstein Equations in general relativity, considered by Thomas Thiemann.
There is proof that this equations exist and theoretically can be solved, but they are very
complicated and until now couldn´t even be solved numerically, which is a big drawback of LQG.
But if well simplified, as done by Martin Bojowald consisting on high symmetries there is a
solution to be found. The simplification depends on a homogeny and isotropic universe which
always looks the same, independent on where you look at it[5].
A collapsing universe (as our universe going backwards in time) is described by the Friedmann
equation
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A Universe BuilT On Loops
An Introduction to Loop Quantum Gravity
Katharina Fließer
̇
with an
̇
,
but if the density of matter energy reaches the Planck scale ̇ vanishes, and the universe stops
contracting. The equations show that it even bounces back and enters in an expanding phase. If
this truly happens no more singularities are needed and the initial singularity problem is solved
the same manner in which the singularity of a pointlike-elektron falling into the infinite coulomb
potential well of a nucleus is resolved by quantum mechanics[4].
So, what are the problems LQG means to solve? The idea is often put forward, that a quantum
theory of gravity has to be a unified theory of all forces. There are hints that this might be the
case, but overriding LQG does not address this problem. The aim of LQG is to fill the lack of
productivity as found in early cosmology, Black Hole Thermodynamics or short-scale structure
of physical space.
Still LQG is just a theory, there are no experiments supporting this (or any other) quantum
theory of gravity. Also, so far, none of those theories has been able to produce a prediction that
could but the theory under a test. The predictions that LQG makes, that for instance, any physical
area must turn out to be quantized, are still far from present measurements. There are also a lot
of other things still missing; LQG is far from being complete. But at least until now there is also
no evidence that LQG is wrong.
[1] Will, C.M., Living Rev.Relativity 9, 3, (2006)
[2] Bojowald, M., Spektrum der Wissenschaft Mai 2009,26 (2009)
[3]Bojowald, M., “Zurück vor den Urknall” (Fischer Verlag, Frankfurt am Main, 2010)
[4]Rovelli, C., Centre de Physique Théorique de Luminy 907, 13288 (2012)
[5]Thiemann T., Pössel M., Spektrum der Wissenschaft Juni 2007, 34 (2007)
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